Electroproduction Structure Functions, Integral Representations, and Light-Cone Commutators

The electroproduction inelastic structure functions Wᵢ(κ, ν) ($i=1, 2$; κ=q²=momentumtransfersquared; ν=q·p=energytransfer; p²=1) are studied in the Bjorken (A) limit (ν→∞, ρ≡-νκ fixed) and in the Regge (R) limit (ν→∞, κ fixed). Finite A limits [F₂(ρ)=limνW₂(κ, ν), F₁(ρ)=limW₁(κ, ν)] and Pomeranchuk-dominated R limits [w²(κ)ν^-1=limW₂, w₁(κ)ν=limW₁] are assumed. These two limits are first related by use of Deser-Gilbert-Sudarshan (DGS) representations for causal functions Vᵢ(κ, ν) related to the Wᵢ by W₂=κV₂ and W₁=κV₁-ν²V₂. The above A- and R-limit assumptions, together with a smoothness assumption on the DGS spectral functions motivated by the existence of some equal-time commutators, are shown to imply that F₂(∞)=w₂(-∞)≡w₂=const and ${lim}_{{ρ}{→}{∞}}{{F}₁({ρ})}{{ρ}}={-}{lim}_{{κ}{→}{-}{∞}}{w}₂({κ}){κ}{≡}{w}₁=const$. These results agree with experiment if ${w}ᵢ{≠}0$. The properties of the spectral functions that are obtained are used to discuss some equal-time commutators and properties of the photon structure function and the cross section ${{σ}}_{{γ}}({ν})$. The empirical value of ${{σ}}_{{γ}}({∞})$ is used to roughly calculate ${w}₂$. It is stressed, however, that the above assumptions do not preclude the possibility that ${w}₂=0$. The Fourier transforms ${{^}{W}}ᵢ({x}², p·{}x)$ of the ${W}ᵢ({κ}, {ν})$ are next studied and used to again relate the $A$ and $R$ limits. Restrictions on the ${W}ᵢ$ imposed by the requirements of finite $A$ limits and correct $R$ limits are determined and results equivalent to the above ones are obtained. This analysis determines the configuration-space behavior corresponding to the $A$ and $R$ limits for large ${ρ}$ and ${κ}$, respectively. These limits are shown to determine the behavior of the ${{^}{W}}ᵢ$ near the light cone ${x}²=0$, and the results are that ${π}{{^}{W}}₂{~}{δ}({x}²)1/2{ε}({x}₀){{^}{f}}₂({x}₀)$ and ${π}{W}₁{~}{{δ}}^{{'}}({x}²)2|{x}₀|{{^}{f}}₁({x}₀)$ for ${x}²{~}0$, where ${w}₂=1/2i{∫}d{x}₀{{^}{f}}₂({x}₀)$ and ${w}₁=2i[{f}₁({∞}){-}{{^}{f}}₁({-}{∞})]$. Corresponding properties of the ${V}ᵢ$ are derived, and the equivalence of this approach with the one using integral representations is established. The light-cone behavior of each component of $〈p|[{J}_{{μ}}(x), {J}_{{ν}}(0)]|p〉$ can be determined, and it is shown in particular that $〈p|[{J}₀{-}{J}₃, {J}₀{-}{J}₃]|p〉{~}{δ}({x}²){ε}({x}₀){{^}{f}}₂({x}₀)$, apart from total derivatives with respect to ${x}₀{-}{x}₃$. The light-cone behavior equivalent to the Fubini-Dashen-Gell-Mann sum rule is then put in a form which can accommodate this result so well that we are led to propose a highly symmetric universal structure for the light-cone commutator of two SU(3) currents. The universality is made precise by use of the [SU(3)SU(3)]_β equal-time commutation relations. The equal-time implications of the proposal are considered and are shown to be consistent with, and suggested by, the gluon mode. In the context of this model, w₂ is numerically estimated and found to agree with the experimental value.